Vector identity proof in general curvilinear coordinates

In summary: Gmj)Gab+δmjΓajbGab+δmjΓajbGab=∂b(Gam)=2∂b(Gmj)Gab+2δmjΓajbGabIn summary, the given equation can be rewritten as ∂b(Gam)=2∂b(Gmj)Gab+2δmjΓajbGab, where ∂b means partial differentiation with respect to b, G is the metric tensor, and Γ is the Christoffel symbol. The key to solving this equation is to use the definitions of the metric tensor and the Christoffel symbol, as well as the product rule for differentiation.
  • #1
Earthland
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Homework Statement



Need to prove that:
phys1.jpg


,b means partial differentation with respect to b, G is the metric tensor and Γ is Christoffel symbol.

I think I could proceed with this quite well if I could understand the hint given, that I should lower the index j.

Homework Equations



am=Gmjaj

The Attempt at a Solution



For the first member on the left I could write

GmjGjkkkmk=∂k

and else than that it should stay the same

For members on the right, lowering j would give me

vα(vj,αjαβvβ)

However, what should I do with the second member on the left? And I am told that this one is the important one.
 
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  • #2


First, let's rewrite the given equation in a more clear form:

∂b(Gam)=∂b(GmjGaj)

Now, we can use the product rule for differentiation to expand the left side:

∂b(Gam)=∂b(Gmj)Gaj+Gmj∂b(Gaj)

Next, we can use the definition of the metric tensor to rewrite the first term on the right side:

∂b(Gam)=∂b(Gmj)Gaj+δmj∂b(Gaj)

Now, we can use the definition of the Christoffel symbol to rewrite the second term on the right side:

∂b(Gam)=∂b(Gmj)Gaj+δmj(∂bGaj+ΓajbGab)

Next, we can use the definition of the metric tensor again to rewrite the first term on the right side:

∂b(Gam)=∂b(Gmj)Gaj+δmj(∂bGaj+ΓajbGab)

=∂b(Gmj)Gaj+δmj(∂b(Gmj)Gab+ΓajbGab)

Now, we can use the product rule again to expand the first term on the right side:

∂b(Gam)=∂b(Gmj)Gaj+δmj(∂b(Gmj)Gab+ΓajbGab)

=∂b(Gmj)Gaj+δmj(∂b(Gmj))Gab+δmjGmj(∂bGab)+δmjΓajbGab

Finally, we can use the definition of the metric tensor one more time to simplify the second term on the right side:

∂b(Gam)=∂b(Gmj)Gaj+δmj(∂b(Gmj))Gab+∂bGab+δmjΓajbGab

=∂b(Gmj)Gaj+δmj(∂b(Gmj))Gab+∂b(Gmj)Gab+δmjΓajbGab

Now, we can see that the first and third terms on the right side are equivalent, and the second and fourth terms are equivalent. Therefore, we can rewrite the equation as:

∂b(Gam)=∂b(Gmj)Gaj+
 

FAQ: Vector identity proof in general curvilinear coordinates

1. What is a vector identity?

A vector identity is a mathematical equation that shows the relationship between different vectors and their components. It is used to simplify and solve problems involving vector quantities.

2. How are vector identities derived in general curvilinear coordinates?

Vector identities in general curvilinear coordinates are derived using tensors and the rules of vector calculus. The components of the vectors are expressed in terms of the coordinate system, and the identities are then derived using differential operators.

3. What is the importance of vector identities in general curvilinear coordinates?

Vector identities in general curvilinear coordinates are important for solving problems in physics and engineering that involve vector quantities in non-Cartesian coordinate systems. They allow for the manipulation and simplification of vector equations in these coordinate systems.

4. Can vector identities be used in any coordinate system?

Yes, vector identities can be used in any coordinate system. However, they are most commonly used in Cartesian, cylindrical, and spherical coordinate systems, as these are the most frequently encountered coordinate systems in physics and engineering.

5. Are vector identities unique to general curvilinear coordinates?

No, vector identities can also be derived and used in Cartesian coordinates. However, vector identities in general curvilinear coordinates are more complex and involve additional terms due to the non-Cartesian nature of the coordinate system.

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